Suppose that $(z_n) \subset \mathbb C$ is a sequence (repetitions allowed) such that $$ F(z) = \prod_n \left ( 1-\frac{z}{z_n} \right ) $$ defines an entire function of exponential type, that is, $|F(z)| \leq ae^{b|z|}$ for $a,b > 0$. Further, let $(p_n) \subseteq \mathbb C$ be a bounded sequence. Does $$ \tilde F(z) = \prod_n \left ( 1-\frac{z}{z_n+p_n} \right ) $$ still define an entire function of exponential type? Notice, that the zeros (z_n+p_n) of the new function $\tilde F$ arise as a "perturbation" of the original zeros $(z_n)$. I feel like the statement is true and should follow from the general theory of entire functions of exponential type but I don't have a reference.

I would appreciate any help.